{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:O2H3EFQHD5EJDJA3PYW55PYMH5","short_pith_number":"pith:O2H3EFQH","schema_version":"1.0","canonical_sha256":"768fb216071f4891a41b7e2ddebf0c3f452f9b0605e35b905308f7cb9e16392f","source":{"kind":"arxiv","id":"1206.4774","version":2},"attestation_state":"computed","paper":{"title":"Arithmetic invariant theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.RT"],"primary_cat":"math.NT","authors_text":"Benedict H. Gross, Manjul Bhargava","submitted_at":"2012-06-21T04:53:06Z","abstract_excerpt":"Let $k$ be a field, let $G$ be a reductive algebraic group over $k$, and let $V$ be a linear representation of $G$. Geometric invariant theory involves the study of the $k$-algebra of $G$-invariant polynomials on $V$, and the relation between these invariants and the $G$-orbits on $V$, usually under the hypothesis that the base field $k$ is algebraically closed. In favorable cases, one can determine the geometric quotient $V//G = Spec(Sym(V^*))^G$ and can identify certain fibers of the morphism $V \\rightarrow V/G$ with certain $G$-orbits on $V$. In this paper, we study the analogous problem wh"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1206.4774","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-06-21T04:53:06Z","cross_cats_sorted":["math.AG","math.RT"],"title_canon_sha256":"e8b7c983c1d1cbb2ab45e4b4fc3365ada1b6f69e83ac9062f8423b9c987b9404","abstract_canon_sha256":"b96386985807f7100c685221a52d2c0ff96814b6922ea220cacedcebe22a37e5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:49:21.664988Z","signature_b64":"DSfNkeD4TzMMP5R+Q5GDrv9qH/8jei6imnHccvGrM4kvrZnJWav5hSA9xfe7/aW3ko3EkP2/J58SEQu5ewzeAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"768fb216071f4891a41b7e2ddebf0c3f452f9b0605e35b905308f7cb9e16392f","last_reissued_at":"2026-05-18T03:49:21.664221Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:49:21.664221Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Arithmetic invariant theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.RT"],"primary_cat":"math.NT","authors_text":"Benedict H. Gross, Manjul Bhargava","submitted_at":"2012-06-21T04:53:06Z","abstract_excerpt":"Let $k$ be a field, let $G$ be a reductive algebraic group over $k$, and let $V$ be a linear representation of $G$. Geometric invariant theory involves the study of the $k$-algebra of $G$-invariant polynomials on $V$, and the relation between these invariants and the $G$-orbits on $V$, usually under the hypothesis that the base field $k$ is algebraically closed. In favorable cases, one can determine the geometric quotient $V//G = Spec(Sym(V^*))^G$ and can identify certain fibers of the morphism $V \\rightarrow V/G$ with certain $G$-orbits on $V$. In this paper, we study the analogous problem wh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.4774","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1206.4774","created_at":"2026-05-18T03:49:21.664341+00:00"},{"alias_kind":"arxiv_version","alias_value":"1206.4774v2","created_at":"2026-05-18T03:49:21.664341+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.4774","created_at":"2026-05-18T03:49:21.664341+00:00"},{"alias_kind":"pith_short_12","alias_value":"O2H3EFQHD5EJ","created_at":"2026-05-18T12:27:16.716162+00:00"},{"alias_kind":"pith_short_16","alias_value":"O2H3EFQHD5EJDJA3","created_at":"2026-05-18T12:27:16.716162+00:00"},{"alias_kind":"pith_short_8","alias_value":"O2H3EFQH","created_at":"2026-05-18T12:27:16.716162+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/O2H3EFQHD5EJDJA3PYW55PYMH5","json":"https://pith.science/pith/O2H3EFQHD5EJDJA3PYW55PYMH5.json","graph_json":"https://pith.science/api/pith-number/O2H3EFQHD5EJDJA3PYW55PYMH5/graph.json","events_json":"https://pith.science/api/pith-number/O2H3EFQHD5EJDJA3PYW55PYMH5/events.json","paper":"https://pith.science/paper/O2H3EFQH"},"agent_actions":{"view_html":"https://pith.science/pith/O2H3EFQHD5EJDJA3PYW55PYMH5","download_json":"https://pith.science/pith/O2H3EFQHD5EJDJA3PYW55PYMH5.json","view_paper":"https://pith.science/paper/O2H3EFQH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1206.4774&json=true","fetch_graph":"https://pith.science/api/pith-number/O2H3EFQHD5EJDJA3PYW55PYMH5/graph.json","fetch_events":"https://pith.science/api/pith-number/O2H3EFQHD5EJDJA3PYW55PYMH5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/O2H3EFQHD5EJDJA3PYW55PYMH5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/O2H3EFQHD5EJDJA3PYW55PYMH5/action/storage_attestation","attest_author":"https://pith.science/pith/O2H3EFQHD5EJDJA3PYW55PYMH5/action/author_attestation","sign_citation":"https://pith.science/pith/O2H3EFQHD5EJDJA3PYW55PYMH5/action/citation_signature","submit_replication":"https://pith.science/pith/O2H3EFQHD5EJDJA3PYW55PYMH5/action/replication_record"}},"created_at":"2026-05-18T03:49:21.664341+00:00","updated_at":"2026-05-18T03:49:21.664341+00:00"}